The Löwner-Kufarev representations for domains with analytic boundaries (Q2890388)

This paper deals with the parametric representation for the class \(\mathcal S\) of all univalent holomorphic functions~\({f:\mathbb D:=\{z\in\mathbb C:|z|<1\}\to\mathbb C}\) normalized by~\(f(0)=0\), \(f'(0)=1\), which is based on the Löwner-Kufarev differential equations NEWLINE\[NEWLINE \frac{\partial F(z,t)}{\partial t}=z\frac{\partial F(z,t)}{\partial z}\,p(z,t),\quad z\in\mathbb D, t\geq0, \eqno{(1)}NEWLINE\]NEWLINE NEWLINE\[NEWLINE \frac{dw(z,t)}{dt}=-w(z,t)\,p\big(w(z,t),t\big),\quad t\geq0, w(z,0)=z\in\mathbb D, \eqno(2) NEWLINE\]NEWLINE where the function \(p:\mathbb D\times[0,+\infty)\to\mathbb C\), called the driving term, is holomorphic in~\(z\), measurable in \(t\), and satisfies the conditions \(\text{Re}\, p>0\) and \(p(0,t)=1\) for all~\(t\geq0\). It is known that for any driving term~\(p(z,t)\) the function \(f\in\mathcal S\) defined by NEWLINE\[NEWLINE f(z):=\lim_{t\to+\infty}e^t w(z,t),\quad z\in\mathbb D,\eqno(3) NEWLINE\]NEWLINE where \(w(z,t)\) stands for the solution to~(2), is the unique (up to linear transformations of~\(\mathbb C\)) initial condition in~(1) at \(t=0\) for which the solution \(F(z,t)\) is well-defined and univalent in~\(\mathbb D\) for all~\(t\geq0\). On the other hand, for any \(f\in\mathcal S\) there exists a (in general non-unique) driving term~\(p(z,t)\) such that \(f\) is given by~(3).NEWLINENEWLINEIn this paper the author studies the case when \(f\) maps \(\mathbb D\) onto a domain bounded by an analytic Jordan curve.NEWLINENEWLINETheorem~1. Let \(f\in\mathcal S\). Then \(f(\mathbb D)\) is a domain bounded by an analytic Jordan curve if and only if there exists a driving term \(p(z,t)\) that generates \(f\) via~(2) and (3) and satisfies \(p(\cdot,t)\equiv1\) for all \(t\geq0\) small enough. For all such \(t\)'s the domains \(F(\mathbb D,t)\) and \(w(\mathbb D,t)\) are also bounded by analytic Jordan curves.NEWLINENEWLINEChoosing a different driving term \(p(z,t)\) one can try to extend the interval~\([0,t_1)\) on the \(t\)-axis for which the latter property of the solutions to~(1) and (2) in Theorem~1 remains true. Assume that \(f\in\mathcal S\) extends to a holomorphic univalent function in a simply connected domain~\(B\supset\overline{\mathbb D}\). Under this condition, in Theorem~2 the author constructs a driving term \(p(z,t)\) generating the function~\(f\) via (2) and (3) for which \(e^{t_1}\) equals the conformal radius of~\(B\) w.r.t.~\(0\).NEWLINENEWLINEIn the next section of the paper, Theorem~3 provides a vast family of solutions to~(1) and (2) in which the functions \(f\in\mathcal S\) and \(w(\cdot,t)\) are univalent quadratic polynomials. However, the elements of the Löwner chain~\(F(\cdot,t)\) are not polynomials in this case.NEWLINENEWLINEFinally, Theorem~4 and Proposition~1 are devoted to the representation of bounded functions in~\(\mathcal S\). Theorem~4 is a restatement of a classical result: If \(f\in\mathcal S\) and \(|f(z)|<M\) for all~\(z\in\mathbb D\), then \(f\) can be represented via~(2) and (3) with a driving term \(p(z,t)\) satisfying \(p(\cdot,t)\equiv1\) for all \(t>\log M\).NEWLINENEWLINEGoing in another direction, Proposition~1 gives a sufficient condition for a bounded univalent function: If there exists \(\beta>0\) such that \(\text{Re}\, p(z,t)>\beta\) for all \(z\in\mathbb D\) and all \(t\geq0\) large enough, then the corresponding function~\(f\) given by~(3) is bounded in~\(\mathbb D\).NEWLINENEWLINEThe author gives an example demonstrating that this condition is not however necessary.